⏳Intro to Time Series Unit 8 Review

A point forecast is a single-number estimate of a future value in a time series. Think of it as your best guess for what will happen at a specific future time point, like predicting next quarter's sales will be 50,000 units.

You generate point forecasts using methods like exponential smoothing or ARIMA models (for example, Holt-Winters or ARIMA(1,1,1)). These concrete estimates drive practical decisions: how much inventory to stock, how many staff to schedule, how to allocate a budget.

Point forecasts also serve two other purposes:

They give you a basis for comparing models. You can evaluate which forecasting method performs best using error metrics like mean squared error (MSE) or mean absolute percentage error (MAPE).
They act as the center point around which you build prediction intervals.

Point forecasts in time series, Better prediction intervals for time series forecasts

Concept of Prediction Intervals

A point forecast alone can be misleading because it hides how confident (or uncertain) you actually are. A prediction interval fixes this by providing a range of values likely to contain the true future observation at a specified probability level.

For example, you might report: "We forecast 125 units, with a 95% prediction interval of [100, 150]." That means you're 95% confident the actual value will fall between 100 and 150.

Key properties of prediction intervals:

They consist of a lower bound and an upper bound centered on the point forecast.
The probability level you choose (80%, 90%, 95%) determines how wide the interval is. Higher confidence means a wider interval.
Wider intervals signal greater uncertainty, while narrower intervals suggest a more precise forecast. Short-term forecasts tend to have narrower intervals than long-term ones, since uncertainty grows over time.

Construction of Prediction Intervals

There are two broad approaches to building prediction intervals: parametric and non-parametric.

Parametric methods assume the forecast errors follow a known distribution, most commonly a normal distribution. Under that assumption, you construct the interval using standard normal quantiles:

$\hat{y}_{T+h} \pm z_{\alpha/2} \times \sigma_h$

where $\hat{y}_{T+h}$ is the point forecast, $z_{\alpha/2}$ is the critical value (e.g., 1.96 for a 95% interval), and $\sigma_h$ is the standard deviation of the forecast errors at horizon $h$ . The interval width is driven entirely by $\sigma_h$ .

Non-parametric methods don't require distributional assumptions:

Empirical quantiles: Sort your historical forecast errors and use the 2.5th and 97.5th percentiles directly (for a 95% interval).
Bootstrap methods: Resample the model's residuals many times, generate many possible future paths, and read off the interval from the resulting distribution of forecasts.

Three factors that influence interval width:

Forecast horizon — Intervals almost always widen as you forecast further ahead. A one-month-ahead interval will be narrower than a one-year-ahead interval because uncertainty accumulates.
Model complexity — A well-specified complex model (like ARIMA) may produce narrower intervals than simple exponential smoothing because it captures more structure in the data. But beware of overfitting: a model that fits noise will produce intervals that are misleadingly narrow.
Variability in the series — A volatile series like daily stock prices will naturally produce wider intervals than a stable series like weekly demand for a household staple.

Uncertainty in Point Forecasts

Prediction intervals exist to remind you (and your stakeholders) that a point forecast is an estimate, not a guarantee. If your point forecast is 120 and your 95% prediction interval is [90, 150], you're saying the true value could plausibly land anywhere in that 60-unit range.

Interpreting prediction intervals carefully:

The probability level you choose is a tradeoff. An 80% interval is narrower and more actionable but less likely to contain the true value. A 95% interval is safer but wider and potentially less useful for tight planning.
Prediction intervals reflect uncertainty within your model. They do not account for structural breaks, regime shifts, or external shocks that your model has never seen. If the economy enters a recession and your model was trained on expansion-era data, the interval may be too narrow.
Rare, extreme events (sometimes called black swan events) can fall outside even a 99% prediction interval. The intervals assume the future broadly resembles the past.

Communicating uncertainty to stakeholders:

Always present prediction intervals alongside point forecasts, whether in a table or a fan chart. A single number without context invites overconfidence.
Explain intervals in plain language: "There's roughly a 95% chance the actual value falls within this range."
Connect uncertainty to decisions. If the lower bound of the interval would trigger a different action than the upper bound, that's a signal for contingency planning or scenario analysis.

⏳Intro to Time Series Unit 8 Review